Coquasi-bialgebras with preantipode and rigid monoidal categories

Abstract

By a theorem of Majid, every monoidal category with a neutral quasi-monoidal functor to finitely-generated and projective $\mathbb{k}$-modules gives rise to a coquasi-bialgebra. We prove that if the category is also rigid, then the associated coquasi-bialgebra admits a preantipode, providing in this way an analogue for coquasi-bialgebras of Ulbrich's reconstruction theorem for Hopf algebras. When $\mathbb{k}$ is field, this allows us to characterize coquasi-Hopf algebras as well in terms of rigidity of finite-dimensional corepresentations.

20 pages, 1 appendix. Mayor changes, improved the presentation, added some further results

References in corpus (2)

• Functorial Constructions for Non-associative Algebras with Applications to Quasi-bialgebras
• On the Structure Theorem for quasi-Hopf bimodules

Cited by in corpus (1)

• Antipodes, preantipodes and Frobenius functors